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NTA Abhyas JEE Main2020MathematicsDifferentiationPractice

If f x = s i n - 1 2 × 3 x 1 + 9 x , then f ' - 1 2 is equal to

Options

  1. A3 l o g e 3
  2. B- 3 l o g e 3
  3. C- 3 l o g e 3
  4. D3 l o g e 3

Correct answer

A. 3 l o g e 3

Step-by-step solution

Given f x = s i n - 1 2 × 3 x 1 + 9 x Let 3 x = t a n t ⇒ t = tan - 1 3 x So, f ( x ) = s i n - 1 2 tan ⁡ t 1 + t a n 2 t We know that, s i n ( 2 t ) = 2 t a n ( t ) 1 + t a n 2 ( t ) ⇒ f ( x ) = s i n - 1 s i n 2 t ∴ f ( x ) = 2 t = 2 t a n - 1 3 x ⇒ d f x d x = 2 1 + ( 3 x ) 2   × 3 x . l o g e 3 At x = 1 2 ,   d f d x = 2 1 + 3 1 2 2 × 3 1 2 . l o g e 3 = 1 2 × 3 × l o g e 3 = 3 × l o g e 3

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